We present a novel constant time algorithm for greyscale (hyper- )spherical flat dilations and erosions. This algorithm is built around our modifications to a recently published fast distance transform for sampled functions. Our method embeds the greyscale image as a binary ``umbra'' in a higher dimensional space and thresholds the distance transform in this new space. The method is: exactly isotropic, time-independent of the structuring function size, and inherently parallelizable at several levels of granularity. Subsequent different size dilations (or erosions) of the same image may also be performed at insignificant further cost. Our testing on a 3D medical image indicates that the method shows advantages for structuring elements with radius greater-than 15 voxels, when compared to some methods from well-known contemporary packages.
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